Implementation of holonomic quantum gates by an isospectral deformation of an Ising dimer chain

We exactly construct one- and two-qubit holonomic quantum gates in terms of isospectral deformations of an Ising model Hamiltonian. A single logical qubit is constructed out of two spin-$\frac{1}{2}$ particles; the qubit is a dimer. We find that the holonomic gates obtained are discrete but dense in the unitary group. Therefore an approximate gate for a desired one can be constructed with arbitrary accuracy.

Figure 1

Dimer consists of two spin-$\frac{1}{2}$ particles.

Figure 2

$\sin {\theta }_{\kappa }$ as a function of $\kappa$.

Figure 3

Rotation angles ${\theta }_{\kappa }=2\kappa \pi ∕\sqrt{3}$ in ${\Gamma }_{(1,0,0)}^{\left(1\right)}\left(\kappa \right)$ are plotted as points $(\cos {\theta }_{\kappa },\sin {\theta }_{\kappa })$ on the unit circle. The numbers in the figure indicate the values of $\kappa$.

Figure 4

Rotation angle $2J$ in the controlled-$e^{i\theta Z}$, ${\Gamma }^C\left(\mathbf{\kappa }\right)$. (a) The value of $2J$ with the unit of $\pi$ is plotted with respect to ${\kappa }_{-}$, when the fixed values of ${\kappa }_{+}$ are given. (b) The values of $\cos 2J$ and $\sin 2J$ are plotted as the points on the unit circle. The plotting marks are the same meaning as in (a).